1,407 research outputs found
Time-fractional diffusion of distributed order
The partial differential equation of Gaussian diffusion is generalized by
using the time-fractional derivative of distributed order between 0 and 1, in
both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general
distribution of time orders we provide the fundamental solution, that is still
a probability density, in terms of an integral of Laplace type. The kernel
depends on the type of the assumed fractional derivative except for the single
order case where the two approaches turn to be equivalent. We consider with
some detail two cases of order distribution: the double-order and the uniformly
distributed order. For these cases we exhibit plots of the corresponding
fundamental solutions and their variance, pointing out the remarkable
difference between the two approaches for small and large times.Comment: 30 pages, 4 figures. International Workshop on Fractional
Differentiation and its Applications (FDA06), 19-21 July 2006, Porto,
Portugal. Journal of Vibration and Control, in press (2007
Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects
We show the asymptotic long-time equivalence of a generic power law waiting
time distribution to the Mittag-Leffler waiting time distribution,
characteristic for a time fractional CTRW. This asymptotic equivalence is
effected by a combination of "rescaling" time and "respeeding" the relevant
renewal process followed by a passage to a limit for which we need a suitable
relation between the parameters of rescaling and respeeding. Turning our
attention to spatially 1-D CTRWs with a generic power law jump distribution,
"rescaling" space can be interpreted as a second kind of "respeeding" which
then, again under a proper relation between the relevant parameters leads in
the limit to the space-time fractional diffusion equation. Finally, we treat
the `time fractional drift" process as a properly scaled limit of the counting
number of a Mittag-Leffler renewal process.Comment: 36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at
the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and
Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006;
Chairmen: R. Klages, G. Radons and I.M. Sokolo
The M-Wright function in time-fractional diffusion processes: a tutorial survey
In the present review we survey the properties of a transcendental function
of the Wright type, nowadays known as M-Wright function, entering as a
probability density in a relevant class of self-similar stochastic processes
that we generally refer to as time-fractional diffusion processes.
Indeed, the master equations governing these processes generalize the
standard diffusion equation by means of time-integral operators interpreted as
derivatives of fractional order. When these generalized diffusion processes are
properly characterized with stationary increments, the M-Wright function is
shown to play the same key role as the Gaussian density in the standard and
fractional Brownian motions. Furthermore, these processes provide stochastic
models suitable for describing phenomena of anomalous diffusion of both slow
and fast type.Comment: 32 pages, 3 figure
Revisiting the derivation of the fractional diffusion equation
The fractional diffusion equation is derived from the master equation of
continuous-time random walks (CTRWs) via a straightforward application of the
Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional
diffusion equation is solved in various important and general cases. The
meaning of the proper diffusion limit for CTRWs is discussed.Comment: Paper presented at the International Workshop on Scaling and
Disordered Systems, Paris, France, 13-14 April 200
Finite Domain Anomalous Spreading Consistent with First and Second Law
After reviewing the problematic behavior of some previously suggested finite
interval spatial operators of the symmetric Riesz type, we create a wish list
leading toward a new spatial operator suitable to use in the space-time
fractional differential equation of anomalous diffusion when the transport of
material is strictly restricted to a bounded domain. Based on recent studies of
wall effects, we introduce a new definition of the spatial operator and
illustrate its favorable characteristics. We provide two numerical methods to
solve the modified space-time fractional differential equation and show
particular results illustrating compliance to our established list of
requirements, most important to the conservation principle and the second law
of thermodynamics.Comment: 14 figure
Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity
In this note we analyze a model for a unidirectional unsteady flow of a
viscous incompressible fluid with time dependent viscosity. A possible way to
take into account such behaviour is to introduce a memory formalism, including
thus the time dependent viscosity by using an integro-differential term and
therefore generalizing the classical equation of a Newtonian viscous fluid. A
possible useful choice, in this framework, is to use a rheology based on
stress/strain relation generalized by fractional calculus modelling. This is a
model that can be used in applied problems, taking into account a power law
time variability of the viscosity coefficient. We find analytic solutions of
initial value problems in an unbounded and bounded domain. Furthermore, we
discuss the explicit solution in a meaningful particular case
Fractional diffusions with time-varying coefficients
This paper is concerned with the fractionalized diffusion equations governing
the law of the fractional Brownian motion . We obtain solutions of
these equations which are probability laws extending that of . Our
analysis is based on McBride fractional operators generalizing the hyper-Bessel
operators and converting their fractional power into
Erd\'elyi--Kober fractional integrals. We study also probabilistic properties
of the r.v.'s whose distributions satisfy space-time fractional equations
involving Caputo and Riesz fractional derivatives. Some results emerging from
the analysis of fractional equations with time-varying coefficients have the
form of distributions of time-changed r.v.'s
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
- …